We prove that if $$\sigma \in S_m$$ σ ∈ S m is a pattern of $$w \in S_n$$ w ∈ S n , then we can express the Schubert polynomial $$\mathfrak {S}_w$$ S w as a monomial times $$\mathfrak {S}_\sigma $$ S σ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar’s orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on $$\mathfrak {S}_w$$ S w being zero-one. In this case, the Schubert polynomial $$\mathfrak {S}_w$$ S w is equal to the integer point transform of a generalized permutahedron.

中文翻译：

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零一舒伯特多项式

我们证明如果 $$\sigma \in S_m$$ σ ∈ S m 是 $$w \in S_n$$ w ∈ S n 的模式，那么我们可以表达舒伯特多项式 $$\mathfrak {S}_w$ $S w 作为单项式乘以 $$\mathfrak {S}_\sigma $$ S σ（在重新索引的变量中）加上具有非负系数的多项式。这意味着舒伯特多项式的所有系数都等于 0 或 1 的排列集在模式包含下是封闭的。使用 Magyar 的正畸，我们通过十二种避免模式的列表来表征这个类别。我们还给出了 $$\mathfrak {S}_w$$ S w 为零一的其他等效条件。在这种情况下，舒伯特多项式 $$\mathfrak {S}_w$$ S w 等于广义排列面体的整数点变换。